Definition of Dot Product and Cross Product

The dot product and cross product are methods of relating two vectors to one another. The dot product is a scalar representation of two vectors, and it is used to find the angle between two vectors in any dimensional space. For vectors and , the dot product is . The cross product is a vector orthogonal to three-dimensional vectors and , and can be used to determine the area or volume of a parallelogram defined by , , and . For vectors and , the cross product is , which is the determinate of a three-by-three matrix. Two vectors are parallel if their cross product is equal to zero, .